Papers

We study simple inhomogeneous deformations of known two-dimensional classical lattice models described by conformal field theory (CFT) at large distances. The deformations are chosen to vary slowly at lattice scales, while preserving critical behavior. Globally, we find that such systems are described by a CFT in curved space, and identify the underlying space metric. Our two examples are the Ising model and the six vertex model with domain wall boundary conditions. In the latter boundaries are also inhomogeneous, which complicates the analysis. We nevertheless solve the free case using hydrodynamics. In the presence of interactions we determine the arctic curve which separates a critical fluctuating region from an ordered (frozen) phase.

We numerically investigate selection criteria for dissipative weak solutions of the compressible Euler equations. Using the Kelvin-Helmholtz problem as a benchmark, we compare four standard numerical schemes - the viscous finite volume (VFV) method, the discrete velocity Boltzmann finite volume (DVBFV) method, the Runge-Kutta discontinuous Galerkin (RKDG) method, and the discontinuous Galerkin spectral element method (DGSEM), and demonstrate that each may converge to a different dissipative weak solution. We evaluate these solutions with respect to several selection criteria based on entropy production, total energy, and energy defect, and examine the sensitivity of selection functionals to the numerical diffusion inherent in each scheme.

We introduce a non-commutative Laplace transform between functionals on path space and formal series in a tensor algebra, under which a natural convolution of path functionals becomes an algebraic product of series. Applying it to a random unitary matrix-valued path development - the quantum path signature - we show that the governing planar loop equations take a non-commutative spectral form. We then extend the loop equations to a $1/N$ genus expansion, organised by topological recursion, and obtain a hierarchy of integral equations on path space for the corrections.

We show that the computational complexity of Reshetikhin-Turaev invariants of simply colored links is preserved when their underlying ribbon fusion categories are related by the zesting construction. Zesting modifies an $A$-graded ribbon fusion category $\mathcal{C}$ with additional algebraic data $\zeta$ to produce a new category $\mathcal{C}^{\zeta}$ whose link invariants are known to differ from those of $\mathcal{C}$ by an invariant of $A$-colored links $\mathcal{J}_{\zeta}$ depending only on $\zeta$. Building on this understanding and on earlier work on quantum braid group representations under zesting, our result suggests how zesting contributes to the organization of (2+1)D topological quantum field theories and topological phases into complexity-theoretic hierarchies. To prove our main result we develop a local formalism analogous to the Reshetikhin-Turaev construction to compute tangle invariants $\mathcal{J}_{\zeta}(T)$, which leads to a polynomial time algorithm to compute invariants of links $\mathcal{J}_{\zeta}(L)$. A byproduct of our construction is an identification (up to a sign) of the link invariants $\mathcal{J}_{\zeta}(L)$ as rack cocycle invariants, which may be of independent interest. Our formalism also extends to define invariants of closed $3$-manifolds with $A$-structure and we obtain similar complexity results for homotopy quantum field theories built from $A$-modular fusion categories.

We study a Purcell three-link microswimmer whose joints are odd-elastic: the torsional stiffness is non-Hermitian, its antisymmetric part $k_o$ injecting mechanical work so that the internal elastic drift is non-conservative. Our main finding is a sharp separation between geometry and cost — odd elasticity is invisible to the control geometry of the swimmer and visible only in the energy of a manoeuvre. The mechanism is a single algebraic fact: the drift lies in the span of the two control vector fields, so the system is feedback-equivalent to a driftless one and the odd modulus enters the bracket structure only through the scalar $\det\textbf{K}=k^2+k_o^2$. From this we deduce that the abnormal extremals of the energy problem are unchanged by $k_o$, that the swimmer is globally controllable for every non-reciprocity with no threshold, and that its nilpotent model is the Cartan $(2,3,5)$ sub-Riemannian structure, deformed only by the metric scaling $g_\chi=(1+\chi^2)g_0$. The odd modulus acts solely on the cost: casting the optimal-steering problem in sub-Finsler (Randers) form, we prove that a prescribed reorientation is strictly cheaper for either sign of $k_o$. Full Resistive-Force-Theory simulations confirm the analysis and reveal that at isotropic drag the swimmer becomes a pure rotator, turning without translating.

This work introduces an algebraic framework yielding explicit, closed matrix differential-difference equations for eighteen models in the exactly solvable sector of the KPZ universality class across four scaling regimes. By organizing Fredholm determinant data into an overdetermined linear problem on a directed lattice graph, we derive a compatibility system termed the diamond equations. Elementary seed data extracted from the shift structure of the Fredholm kernel provides simple solutions to this system. We then construct a Darboux transformation to compress the infinite-dimensional Fredholm data into a finite-dimensional matrix observable. We show this dressing procedure preserves the diamond equations; consequently the resulting matrix observable obeys the same nonlinear structure as the initial seed data. Verifying a closed nonlinear equation for any specific model thus reduces to checking a handful of linear conditions on its kernel data. Under a scalar reduction, the framework produces variable-coefficient Hirota-Miwa equations for Fredholm determinants, recovering the one-point bilinear equations of the author's earlier work as specializations. To supply the necessary seed data, a product graph construction with admissible propagators builds multipoint data in the fully discrete regime, while Euclidean division in a polynomial quotient algebra handles vertex and polymer models. Finally, we demonstrate the diamond equations are a gauge-equivalent reparametrization of the non-abelian Hirota-Miwa system, a central system in classical integrability theory.

Spectral transitions in non-Hermitian lattices often arise from the competition between non-reciprocal skin accumulation and inter-component hybridization. In short-range systems formed by two coupled chains, this competition conventionally leads to the logarithmic critical-length law $N_c\sim\ln D$, where $D$ is the transverse separation between the chains. Here we show that long-range hoppings fundamentally reorganizes this critical behavior, producing a hierarchy of distinct scaling laws. When only the hybridization couplings are power-law decaying with exponent $\alpha$, the onset becomes algebraic, $N_c\sim D^{\alpha/3}$. When the hoppings within each chain are themselves also power-law decaying, in addition to the hybridization couplings, the system enters a scale-covariant regime for $\alpha<2$, in which the criticality threshold equation depends only on the system aspect ratio $N_c/D$. At $\alpha=2$ and beyond, this regime is followed by a marginal logarithmically corrected and algebraically corrected regimes, respectively. We identify two new non-local mechanisms that enable this unconventional critical hierarchy: a nonanalytic band-edge dispersion from long-range intra-chain hoppings, and parity-mixing hybridization induced by non-reciprocity. Our results show that nonlocality systematically removes the physical length scales i.e. skin depth underlying conventional critical non-Hermitian skin behavior, offering a platform-independent framework testable in programmable topoelectrical circuits, photonic lattices and digital quantum simulators.

Universal hallmarks of quantum chaos—such as the ramp and plateau in the spectral form factor—and the ramp's gravitational duals involving wormholes are widely interpreted as consequences of spectral or ensemble averaging. In this paper, following an earlier proposal of~\cite{Liu25c}, we develop an alternative approach: these phenomena arise as macroscopic smooth structures hidden within erratic microscopic data, which can be isolated through a smooth filter projection. Using the semiclassical Gutzwiller trace formula as a paradigmatic example, we illustrate how many features characteristic of random matrix models—including the ramp, the plateau, the spectral curve, and single-eigenvalue instantons—can be derived in the semiclassical limit without invoking ensemble or explicit spectral averages. We postulate the existence of a minimal Gutzwiller-like structure in the large-$N$ limit of holographic systems and explore its consequences. Beyond deriving the ramp and the plateau, this Gutzwiller-like structure predicts universal rapid macroscopic oscillations in the density of states and the possible existence of hyper-instantons, both of which involve double exponentials in $1/N^2$. On the gravity side, we demonstrate how spacetime wormholes enable the construction of emergent hyper-non-perturbative objects—such as baby-universe and wormhole condensates—which yield double exponential effects in $G_N$. This mirrors the postulated boundary Gutzwiller-like structure and provides a dual gravitational derivation of the universal rapid macroscopic oscillations in the density of states and the spectral plateau.

We study the BRST quantization of the null string after completing its local gauge symmetry by Carroll-Weyl transformations. The resulting worldsheet theory possesses three first-class constraints, $C_1 = P^2$, $C_2 = P \cdot X'$, and $C_3 = P \cdot X$, whose modes realize a Weyl-BMS algebra. The additional Carroll-Weyl constraint qualitatively changes the quantum gauge complex: its scalar $s$-ghost is intrinsically coupled to the BMS $bc$-ghost sector, and the anomaly analysis involves three independent cocycles rather than a single Virasoro-type central charge. Starting from the gauge-fixed action, we derive the complete Faddeev-Popov complex, construct the matter and ghost currents and the BRST charge, and evaluate their equal-time operator products in the flipped, equivalently highest-weight, representation. The matter and ghost anomaly coefficients are $(c_{LL},c_{LS},c_{SS})_{\mathrm{matter}}=(2D,-D,-D)$ and $(c_{LL},c_{LS},c_{SS})_{\mathrm{ghost}}=(-54,6,4)$. Because the corresponding central terms multiply linearly independent ghost bilinears in $Q_B^2$, BRST nilpotency requires the three conditions $D=27$, $D=6$, and $D=4$, respectively. These conditions are mutually incompatible. Consequently, there is no target-space dimension in which the minimal flat Carroll-Weyl matter-plus-ghost complex is anomaly-free in the highest-weight representation. The familiar $D=26$ condition of the ILST null string is recovered only after truncation to the two-constraint BMS subsector, which defines a different quantum gauge complex.

For integers $N\ge0$ and $n\ge2$, let \[ \Delta_N^{(n)} =\left\{\alpha\in\mathbb Z_{\ge 0}^n: \alpha_1+\cdots+\alpha_n=N\right\}. \] We study discrete unique continuation for functions on this lattice simplex. For an integer $R\ge1$, let $g:\Delta_{nR}^{(n)}\to\mathbb R$ satisfy \[ \sum_{i=1}^n g(\beta+e_i)=0, \qquad \beta\in\Delta_{nR-1}^{(n)}, \] where $e_i$ is the $i$th standard basis vector. We prove that nonvanishing at the center implies \[ |\operatorname{supp} g|\ge c_n R^{\lceil n/2\rceil}. \] Here $\operatorname{supp} g$ is the set of points where $g$ is nonzero, and $c_n>0$ depends only on $n$. The exponent $\lceil n/2\rceil$ is optimal. The proof represents the values of $g$ as polynomial coefficients and uses a Pascal uncertainty principle, which bounds from below the total number of nonzero coefficients of a one-variable polynomial and its unit translate in terms of their degree.

Analytic calculations of zero- and single-scale quantities in perturbative quantum field theory result into special numbers and functions, the first of which have been revealed during the last decades. These are generalizations of the polylogarithm in form of Kummer-Poincaré iterative integrals over special alphabets and extensions thereof.With growing order in the coupling constant, the polylogarithms, Nielsen integrals, the iterated integrals over linear denominator terms, cyclotomic letters, letters induced by quadratic forms, square-root valued letters, and more general functions contribute. For the nested sums we consider nested harmonic sums, generalized harmonic sums, nested sums implied by quadratic forms, cyclotomic harmonic sums, and nested sums containing central binomials. We construct the $q$-extensions of these special functions and of the nested sums, which are associated to them by the series expansion at $x=0$, and their Mellin transform in the $q$-free case. These functions are expected to play a role in perturbative calculations in the case of $q$-deformed commutation relations. For the simpler function spaces closed form solutions are presented. For more involved alphabets we present the algorithmic steps leading to the $q$-extension for the individual cases. We also derive the determining differential and difference equations of these higher transcendental functions. The $q$-extended special functions are quite different form the corresponding $\mu$-extended functions.

We construct an autonomous Lipschitz fast dynamo on the flat three-torus whose particle flow has zero topological entropy and whose ideal induction equation has no exponential growth. For every sufficiently small positive magnetic diffusivity, the induction operator has an eigenvalue with real part bounded below by a positive constant independent of the diffusivity. A corresponding non-zero real-valued magnetic field satisfies an exact exponential growth law in $\mathrm{L}^2$. For the same velocity field, every ideal solution grows at most linearly in $\mathrm{L}^2$. Moreover, the particle flow and its inverse have Lipschitz constants growing at most linearly in time. The velocity field is real-valued, divergence-free, differentiable everywhere and smooth away from one circle, but is not $\mathrm{C}^1$.

We study the dependence on time of the spatial regularity of solutions to the periodic Benjamin–Ono equation with real, mean-zero initial data of bounded variation. For every datum with a nonzero jump, we prove that continuity and essential boundedness at an irrational time $t=2\pi\alpha$ are equivalent to the arithmetic condition $\sum_j\log(q_{j+1})/\sqrt{q_j}<\infty$, where $q_j$ are the continued-fraction denominators of $\alpha$. We also determine the exact Hölder regularity and the Besov scale for spatial exponents $2<p\leq\infty$. We derive a high-frequency energy formula and prove the same sharp local criteria on every spatial interval at every irrational time for an open dense set of BV data. At rational times, piecewise smooth data give finite sums of logarithmic and jump kernels modulo a continuous remainder. For arbitrary BV data with a jump, the nonlinear and linear solutions have different logarithmic Poisson coefficients at every sufficiently large reduced denominator, uniformly in the numerator. The proofs combine the smoothing of Tao's gauge transform with estimates for quadratic Fourier series. A finite averaging argument isolates an atom of the initial gauge measure while retaining the real nonlinear reconstruction.

Imaginary Liouville theory was recently proposed as a path integral construction of a non-rational conformal field theory (CFT) with central charge less than one (Usciati et al. 2026). In the present work, we establish Ward identities and Belavin-Polyakov-Zamolodchikov differential equations in this framework, assuming a precise conjecture on the decay of the Laplace transform of Imaginary Gaussian multiplicative chaos. These results provide a first step towards a rigorous implementation of the conformal bootstrap program for conformal field theories with central charge lower than one. This work is the first to investigate the intrinsic properties of Imaginary Liouville theory beyond exact computations.

The Poisson-Boltzmann (PB) theory is a cornerstone of implicit solvent models for electrostatic analysis, and has found a great success in various biomolecular applications. However, in calculating polar solvation energy, one should consider that the structure of the protein changes upon transition from vacuum to water phases. To address this, here we report for the first time a generalized PB framework capable of accommodating nonrigid conformational changes without suffering from self-energy artifacts. For regularized PB models, in which the charge singularities are captured by the Green's functions, self-energies in the water and vacuum states will be analytically canceled. For non-regularized PB solvers, such as APBS and DelPhi, a simple thermodynamic cycle is proposed for nonrigid proteins by adding a Coulombic correction in vacuum. The generalized PB theory is validated using a perturbed two-atom system and a diverse set of proteins with different structures in vacuum and water, demonstrating its accuracy and robustness, regardless of the choice of sharp-interface and diffuse-interface PB models and different numerical solvers.

We develop a perturbative response calculus for spectral problems obtained by changing the speed measure of a fixed symmetric energy form in a Gaussian environment. If Cameron–Martin translation acts by $\mu_{x+f}=e^{\kappa f}\mu_x$, unitary transport identifies the varying $L^2$ spaces and produces a common-domain analytic family. For a positive eigenvalue $\Lambda$, the first-order operator is $-\kappa\Lambda$ times the compression of multiplication by $f$ to the $\Lambda$-eigenspace; its eigenvalues are the derivatives of the analytic branches issuing from $\Lambda$. A countable separation condition and finite-dimensional Gaussian disintegration then give almost-sure simplicity; a response-transversality condition and the inverse function theorem give joint densities for all finite vectors of positive ordered eigenvalues. We verify these hypotheses for every $0<\gamma<2$ in two models: Dirichlet Liouville Brownian motion on an arbitrary bounded connected planar domain, and the Liouville–Cauchy operator on the circle. In the first model the whole spectrum is almost surely simple; in the second the constants form the deterministic zero mode and the positive spectrum is almost surely simple. Transversality follows from a local eigenfunction-square identity in the Brownian case and its nonlocal jump-form analogue in the Cauchy case.

We investigate the quantum interference and energetic phenomena associated with classical caustics in the quantum bouncing ball problem, we refer to as quantum caustics. By considering an initial Gaussian wavepacket, we show that caustics associated with the underlying classical trajectory families are exhibited. We connect the associated phase singularity chains in the vicinity of the caustic with the semiclassical Pearcey function built on the cusp catastrophe lines. We also quantify the amount of superenergy exhibited in these solutions - regions of space where the local energy exceeds the largest constituent energy eigenvalue. We give a complimentary description of the caustic and superenergy behavior using the Madelung/Bohm trajectories, which gives additional insight about the energy of the trajectories and how they traverse the phase singularity chains.

Preserving metrological information under noise is central to quantum sensing, yet finite detectors and memories impose an unavoidable limit on how finely noise trajectories can be resolved. Using exact symmetric-logarithmic-derivative geometry, we determine when a monitored trajectory can be compressed without losing quantum Fisher information. For an explicit faithful monitored qubit family, the complete joint signal-and-noise model is recoverable from only polynomially many type records, requiring $O(\log n)$ terminal memory, whereas deferred recovery of arbitrary $n$-qubit states requires exponentially many trajectories, or $O(n)$ memory. Online correction replaces this terminal storage by an irreducible per-use readout and feedback alphabet. These results establish syndrome information as a task- and timing-dependent resource connecting quantum sensing, statistical sufficiency, and quantum error correction.

We formulate the loop equations of pure $4d, N=1$ super Yang–Mills (SYM) theory in a finite geometric form. The usual equal-point loop derivatives are singular because the order of gauge-field insertions is lost when contour points coincide. Our path-ordered operator calculus (POOC) keeps the insertions in distinct ordered slots, forms the graded commutators and Jacobi combinations, and only then takes the coincidence limit. This removes the spurious kinematical singularities without introducing a cutoff; the genuine ultraviolet contact remains a separate physical distribution. We apply POOC to the exact Itoyama–Takashino superloop hierarchy and construct a Lorentzian supersymmetric Hodge-dual (SHD) surface functional. Its Hodge-resolved area derivative is annihilated by the local chiral and anti-chiral loop operators. Exact additivity under surface-geodesic sewing shows that the exponential of this zero mode multiplies any solution of the complete finite-$N$ hierarchy without changing the equations. For planar contours the SHD functional is the geometric area. In particular, for a long rectangular Wilson loop, $W[C_{T,L}]\sim\exp[-i\sigma_kLT]$ and $E_k(L)=\sigma_kL>0$. Thus the construction gives an exact nonperturbative confining area-law factor in $ N=1$ SYM. After Euclidean continuation, the same SHD factor is an exact fixed point of the deterministic zero-noise SYM gradient flow. Its interpretation as a dynamically selected equilibrium still requires stability in the long-flow-time, large-volume, and zero-noise limits, but this does not affect the exact zero-mode and finite-$N$ dressing theorems. The complete planar Wilson-loop solution, including the undressed fluctuation factor and excitation spectrum, will be developed in the next papers of this series.

In this paper, we establish the exponential convergence of Laskar's pioneering frequency map analysis, strictly improving upon classical polynomial bounds. By utilizing appropriately chosen weighting functions, we achieve such exponential rates in the analytic quasi-periodic regime for frequency vectors satisfying Diophantine or Brjuno nonresonance conditions. Furthermore, we extend the theoretical framework beyond the analytic quasi-periodic regime into the analytic almost periodic setting. By employing and generalizing Bourgain's framework, we accommodate a much broader class of anisotropic spatial structures. These results yield the first unified theory of exponential convergence for frequency map analysis, revealing the interplay among analyticity, spatial structures, nonresonance conditions, the choice of weighting functions, and convergence rates. This framework has important implications for applications in fields including celestial mechanics. Moreover, the novel techniques developed herein yield new insights into the study of weighted Birkhoff averages.

This work investigates the $q$-deformation of $(r,s)$-Airy structures and their realization via $q$-difference operators, providing a bridge between quantum spectral curves and integrable systems. We construct an all-order $q$-WKB solution for the matrix systems associated with the $q$-quantized curve $E_q(x,y)=0$. We demonstrate that the resulting non-perturbative connected $q$-amplitudes satisfy a set of shifted $q$-loop equations, which can be interpreted as the Ward identities of a $q$-deformed $\mathcal{W}(\mathfrak{gl}_r)$ algebra. Our main result provides a rigorous classification of admissible $(r,s,q)$ pairs and $q$-Casimir configurations that satisfy the $q$-topological type property. This ensures that the semi-classical expansion is uniquely governed by the $q$-topological recursion, offering new insights into the $q$-quantization of mirror curves and their underlying algebraic structures.

We investigate the two-dimensional inhomogeneous Ising model (2DIM) on the kagom'e lattice by mapping it onto a particular non-symmetric eight-vertex model and constructing the corresponding $R$-matrix. Using a fermionic representation, we evaluate the partition function and derive explicit expressions for the main thermodynamic quantities. In the thermodynamic limit, we obtain an exact equation for the critical surface determining the phase transition of the model. We also calculate the free energy, specific heat, and spontaneous magnetization in the ferromagnetic case. Furthermore, we show that when one or two coupling constants vanish, the model reduces, respectively, to the square-lattice and one-dimensional Ising models. In both limits, our results reproduce the corresponding exact critical couplings and free energies.

Here, we investigate the motion of a thin circular quantum vortex filament near the infinite planar surface. The fluid surrounding this surface moves with a non-zero velocity $\bf{v}$, which is parallel to the surface. We study the specific features of this quantum system and show that they are quite suitable for the boundary layer theory. The developed model allows us to calculate the vortex energy spectrum, $E = E({\bf p})$, where ${\bf p}$ is the total momentum of a vortex ring. We have demonstrated that this function has complex non-trivial dependence on the velocity $\bf{v}$. It is stated that the inverse effective mass of a vortex under consideration is of a tensorial nature. In certain quantum states, the system shows the possibility of both negative and positive effective mass existing. This study employs a novel quantization method for classical closed vortex filaments, developed by the author earlier.

A model governing quasi-unidirectional wave propagation at the surface of a shallow layer of water is derived from Euler equations by long wave asymptotics together with Hamiltonian reduction techniques. The balance between nonlinearity, dispersion and weak dependence on the second planar coordinate is examined and results in the so-called CH-KP equation, a mildly nonlinear version of the well known Kadomtsev-Petviashvili equation. The Hamiltonian structure of the model is obtained by Dirac reduction from the intermediate step of a fully-nonlinear, long-wave system for stratified fluids. A particular scaling is considered that reduces such system to the one-dimensional CH equation in a limiting case. Examples of weak peakon-type solutions are provided, chosen to illustrate the collision behavior supported by the limiting equation.

Generalised continuum theories couple the macroscopic deformation and the micro-/meso-scopic deformation of an underlying micro-structure. They can account for internal length-scale effects and higher-order mechanical loadings absent from classical Cauchy elasticity and has shown its efficiency in modelling metamaterials. This has led to a proliferation of higher-grade and higher-order models (e.g. strain-gradient, micromorphic, micro-polar) whose underlying kinematic reduction strategies, in particular to reduce the number of material parameters, are rarely identifiable from the free energy alone. It leaves two open issues: the lack of a systematic criterion for selecting an appropriate model, and a persistent ambiguity regarding the physical status of the local frames (”directors of matter”) used in their kinematic description. Are they physical quantities describing the change of state of the given micro-structure or arbitrary kinematic descriptors of its deformation\,? This work addresses both questions through a gauge-theoretic formulation of generalised continua in finite strains. While relying on tools and modelling choices consistent with the existing geometric literature on continuum mechanics, the present approach departs from it in its objectives, aiming at a unifying classification of available mechanical models rather than the description of a specific microstructural phenomenon such as defects. In this work, generalised configurations are defined as moving frames over classical configurations, and invariance with respect to the reference generalised configuration is shown to be a necessary and sufficient condition for a gauge invariance, recovering the micromorphic theory as the general-purpose theory of the deformation of arbitrary directors of matter. A systematic classification of first-order generalised media follows from structural group reduction, while strain-gradient continua are recovered through convected frames, and finally constrained media (e.g. couple-stress) are addressed.

We prove that the entanglement of formation of every two-mode Gaussian state coincides with its Gaussian restriction, solving a longstanding open problem in continuous variable quantum information theory. The key result is a sharp affine relation between entanglement and a generalized Einstein–Podolsky–Rosen observable, valid for arbitrary pure two-mode states, including non-Gaussian ones. We further prove that the entanglement of formation is additive on arbitrary tensor products of two-mode Gaussian states, and hence coincides with the entanglement cost, settling the latter for all two-mode Gaussian states. As a consequence, we obtain in closed form the entanglement cost of thermal-loss, thermal-amplifier and additive-noise bosonic channels. Our results, which also extend to bisymmetric multimode states, establish that no two-mode state is less entangled than the Gaussian state with the same covariance matrix, yielding measurable lower bounds on both the entanglement of formation and the entanglement cost of arbitrary non-Gaussian states.

We present a novel approach to the superintegrability of collective Hamiltonians invariant under a Hamiltonian action of a connected semisimple compact Lie group, $G$, on a symplectic manifold, $M$. By exploiting a Hamiltonian torus action that goes back to Guillemin and Sternberg [GS,1983], we demonstrate that the functional dimensions of $\mathfrak{H} := \mathcal{J}^*(C^\infty(\mathfrak{g}^*)^G)$, where $\mathcal{J}: M \to \mathfrak{g}^*$ is the momentum map of the $G$ action, and its centralizer $\mathfrak{F}$ in $C^\infty(M)$ satisfy the equality $\mathrm{ddim}(\mathfrak{H}) + \mathrm{ddim}(\mathfrak{F}) = \mathrm{dim}(M)$. Together with a non-triviality condition, this ensures that the Abelian Poisson algebra $\mathfrak{H}\subset C^\infty(M)$ represents a superintegrable system, and it also follows that the momentum map of the GS torus action yields action variables for the system. Our work provides a new insight into collective superintegrability complementing earlier results of Bolsinov and Jovanović.

We propose a quantum dominant energy condition (QDEC) for the stress tensor in the context of quantum field theory in curved spacetimes. A rigorous proof is given for the case of Rindler wedges, and a heuristic discussion about possible generalizations to more general geometric setups, including curved spacetime, is provided. In Minkowski spacetime, we establish a connection with state recovery bounds. We illustrate the QDEC for coherent states of the free scalar field, where it turns out to be related to the ordinary DEC for the stress tensor of the classical solution that defines the coherent state.

We study the Katz-Long-Moody (KLM) construction and classify exactly when the resulting braid group representations factor through Hecke algebras, for scalar braid part and semisimple free-group part, over an algebraically closed field of characteristic zero and at every convolution parameter lambda different from 1. We show that, except for one exceptional two-strand family, this property then depends only on the eigenvalues of the free-group part and is independent of lambda. Semisimplicity is a genuine hypothesis: we exhibit non-semisimple inputs, namely g = I + N with N nonzero and N^2 = 0, whose KLM quotient is nonetheless a Hecke module, realizing the permutation representation of the symmetric group. We also give an exact criterion for when the resulting representations factor through Temperley-Lieb algebras.

We obtain closed explicit formulas for the spectral determinants of the smooth hyperbolic Bolza surface and the Klein quartic. In each case, a multiplicative relation expresses the determinant of the surface in terms of determinants of singular quotient orbifolds of genera zero and one. The elliptic factors are evaluated by applying the singular Polyakov anomaly formula to explicit Belyi maps on CM elliptic curves of discriminants -8 and -7, while the genus-zero factors are evaluated by explicit determinant formulas for constant-curvature spheres with conical singularities. The same multiplicative relations hold fibrewise on the corresponding equisymmetric deformation strata and yield determinant and first-variation identities. We also prove that every compact quasiplatonic hyperbolic surface is a critical point of the spectral determinant on its Teichmüller space; in particular, this applies to the Bolza surface and the Klein quartic.